LMFDB - Newform orbit 88.3.h.a

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [88,3,Mod(65,88)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(88, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("88.65");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 88 = 2^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	88.h (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$2.39782632637$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.0.1750426112.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 21x^{4} + 4x^{3} + 228x^{2} + 368x + 548 $ x^6 - 2x^5 - 21x^4 + 4x^3 + 228x^2 + 368*x + 548
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $q$-expansion are expressed in terms of a basis $1,\beta_1,\ldots,\beta_{5}$ for the coefficient ring described below. We also show the integral $q$-expansion of the trace form.

$f(q)$	$=$	$ q + ( - \beta_1 - 1) q^{3} + \beta_{3} q^{5} - \beta_{2} q^{7} + (\beta_{3} + 1) q^{9} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 2) q^{11} + (\beta_{5} - 2 \beta_{4} + \beta_{2}) q^{13} + ( - 2 \beta_{3} - 3 \beta_1 - 1) q^{15}+ \cdots + (4 \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 33) q^{99}+O(q^{100}) $ q + (-b1 - 1) * q^3 + b3 * q^5 - b2 * q^7 + (b3 + 1) * q^9 + (b4 + b3 - b2 + b1 + 2) * q^11 + (b5 - 2b4 + b2) q^13 + (-2b3 - 3b1 - 1) * q^15 + (-b5 - b2) * q^17 + (2b4 + b2) q^19 + (-b5 + 3b2) q^21 + (-b1 - 9) * q^23 + (-3b3 + 4b1 + 5) * q^25 + (-2b3 + 5b1 + 7) * q^27 - 2b4 q^29 + (-4b3 + 3b1 - 5) * q^31 + (b5 + 2b4 - 3b3 + b2 - 4b1 - 12) q^33 + (4b5 - 4b4 + b2) * q^35 + (5b3 + 12b1 - 4) * q^37 + (-4b5 + b2) q^39 + (b5 + 4b4 - 3b2) * q^41 + (-4b5 + 2b4 - 2b2) q^43 + (-2b3 + 4b1 + 30) * q^45 + (10b3 - 6b1 - 12) * q^47 + (-4b3 - 20b1 - 35) * q^49 + (-4b4 + 3b2) * q^51 + (4b3 - 12b1 + 14) * q^53 + (4b5 - 2b4 + 3b2 + 7b1 + 31) * q^55 + (5b5 + 4b4 - 7b2) q^57 + (4b3 - 13b1 + 35) * q^59 + (-6b5 - 2b4 + 2b2) q^61 + (4b5 - 4b4) * q^63 + (-5b5 + 4b4 - 13b2) q^65 + (8b3 - b1 - 49) q^67 + (b3 + 8b1 + 18) q^69 + (8b3 + 15b1 + 55) * q^71 + (-b5 + 4b4 + 3b2) * q^73 + (2b3 + 8b1 - 38) * q^75 + (5b5 - 4b4 + 4b3 - 3b2 - 12b1 - 44) q^77 + (4b5 + 8b4 - 4b2) q^79 + (-10b3 + 4b1 - 59) * q^81 + (-8b5 - 2b4) * q^83 + (5b5 - 8b4 + 9b2) q^85 + (-4b5 - 4b4 + 4b2) q^87 + (-3b3 - 12b1 + 60) * q^89 + (-28b3 - 12b1 - 12) * q^91 + (5b3 + 20b1 - 18) * q^93 + (-4b5 + 8b4 + 3b2) q^95 + (-7b3 + 16b1 + 72) * q^97 + (4b5 - b4 + b3 + 2b2 + 8b1 + 33) q^99
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q - 4 q^{3} + 6 q^{9} + 10 q^{11} - 52 q^{23} + 22 q^{25} + 32 q^{27} - 36 q^{31} - 64 q^{33} - 48 q^{37} + 172 q^{45} - 60 q^{47} - 170 q^{49} + 108 q^{53} + 172 q^{55} + 236 q^{59} - 292 q^{67} + 92 q^{69}+ \cdots + 182 q^{99}+O(q^{100}) $ 6 * q - 4 * q^3 + 6 * q^9 + 10 * q^11 - 52 * q^23 + 22 * q^25 + 32 * q^27 - 36 * q^31 - 64 * q^33 - 48 * q^37 + 172 * q^45 - 60 * q^47 - 170 * q^49 + 108 * q^53 + 172 * q^55 + 236 * q^59 - 292 * q^67 + 92 * q^69 + 300 * q^71 - 244 * q^75 - 240 * q^77 - 362 * q^81 + 384 * q^89 - 48 * q^91 - 148 * q^93 + 400 * q^97 + 182 * q^99

Basis of coefficient ring in terms of a root $\nu$ of $ x^{6} - 2x^{5} - 21x^{4} + 4x^{3} + 228x^{2} + 368x + 548 $ x^6 - 2*x^5 - 21*x^4 + 4*x^3 + 228*x^2 + 368*x + 548 :

$\beta_{1}$	$=$	$ ( 137\nu^{5} - 977\nu^{4} - 1209\nu^{3} + 15913\nu^{2} + 22818\nu - 82318 ) / 28204 $ (137v^5 - 977v^4 - 1209v^3 + 15913v^2 + 22818*v - 82318) / 28204
$\beta_{2}$	$=$	$ ( 35\nu^{5} - 404\nu^{4} - 103\nu^{3} + 720\nu^{2} + 11182\nu + 8512 ) / 7051 $ (35v^5 - 404v^4 - 103v^3 + 720v^2 + 11182*v + 8512) / 7051
$\beta_{3}$	$=$	$ ( -405\nu^{5} + 1653\nu^{4} + 5221\nu^{3} - 17397\nu^{2} - 24634\nu + 218 ) / 28204 $ (-405v^5 + 1653v^4 + 5221v^3 - 17397v^2 - 24634*v + 218) / 28204
$\beta_{4}$	$=$	$ ( 435\nu^{5} - 992\nu^{4} - 11353\nu^{3} + 6934\nu^{2} + 130918\nu + 133996 ) / 14102 $ (435v^5 - 992v^4 - 11353v^3 + 6934v^2 + 130918*v + 133996) / 14102
$\beta_{5}$	$=$	$ ( 268\nu^{5} - 676\nu^{4} - 4012\nu^{3} + 1484\nu^{2} + 58224\nu + 53896 ) / 7051 $ (268v^5 - 676v^4 - 4012v^3 + 1484v^2 + 58224*v + 53896) / 7051

Display $\nu^j$ in terms of $\beta_i$

$\nu$	$=$	$ ( \beta_{5} + 4\beta_{3} + 4\beta _1 + 4 ) / 8 $ (b5 + 4b3 + 4b1 + 4) / 8
$\nu^{2}$	$=$	$ ( \beta_{3} - 2\beta_{2} + 5\beta _1 + 17 ) / 2 $ (b3 - 2b2 + 5b1 + 17) / 2
$\nu^{3}$	$=$	$ ( 7\beta_{5} - 6\beta_{4} + 9\beta_{3} - 3\beta_{2} + 13\beta _1 + 45 ) / 2 $ (7b5 - 6b4 + 9b3 - 3b2 + 13*b1 + 45) / 2
$\nu^{4}$	$=$	$ ( 18\beta_{5} - 8\beta_{4} + 25\beta_{3} - 52\beta_{2} + 37\beta _1 + 109 ) / 2 $ (18b5 - 8b4 + 25b3 - 52b2 + 37*b1 + 109) / 2
$\nu^{5}$	$=$	$ ( 297\beta_{5} - 220\beta_{4} - 50\beta_{3} - 330\beta_{2} + 86\beta _1 + 470 ) / 4 $ (297b5 - 220b4 - 50b3 - 330b2 + 86*b1 + 470) / 4

Display $\beta_i$ in terms of $\nu^j$

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/88\mathbb{Z}\right)^\times$.

$n$	$23$	$45$	$57$
$\chi(n)$	$1$	$1$	$-1$

Embeddings

For each embedding $\iota_m$ of the coefficient field, the values $\iota_m(a_n)$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label

$\iota_m(\nu)$

$ a_{2} $

$ a_{3} $

$ a_{4} $

$ a_{5} $

$ a_{6} $

$ a_{7} $

$ a_{8} $

$ a_{9} $

$ a_{10} $

65.1

4.50331	−	1.41421i
4.50331	+	1.41421i
−2.88824	+	1.41421i
−2.88824	−	1.41421i
−0.615072	+	1.41421i
−0.615072	−	1.41421i

−3.88824

5.11838

−

12.7373i

6.11838

65.2

−3.88824

5.11838

12.7373i

6.11838

65.3

−1.61507

−7.39154

−

8.16917i

−6.39154

65.4

−1.61507

−7.39154

8.16917i

−6.39154

65.5

3.50331

2.27316

−

1.73969i

3.27316

65.6

3.50331

2.27316

1.73969i

3.27316

Inner twists

Char	Parity	Ord	Mult	Type
1.a	even	1	1	trivial
11.b	odd	2	1	inner

Twists

By twisting character orbit
Char	Parity	Ord	Mult	Type	Twist	Min	Dim
1.a	even	1	1	trivial	88.3.h.a	✓	6
3.b	odd	2	1		792.3.j.a		6
4.b	odd	2	1		176.3.h.d		6
8.b	even	2	1		704.3.h.h		6
8.d	odd	2	1		704.3.h.g		6
11.b	odd	2	1	inner	88.3.h.a	✓	6
12.b	even	2	1		1584.3.j.k		6
33.d	even	2	1		792.3.j.a		6
44.c	even	2	1		176.3.h.d		6
88.b	odd	2	1		704.3.h.h		6
88.g	even	2	1		704.3.h.g		6
132.d	odd	2	1		1584.3.j.k		6

By twisted newform orbit
Twist	Min	Dim	Char	Parity	Ord	Mult	Type
88.3.h.a	✓	6	1.a	even	1	1	trivial
88.3.h.a	✓	6	11.b	odd	2	1	inner
176.3.h.d		6	4.b	odd	2	1
176.3.h.d		6	44.c	even	2	1
704.3.h.g		6	8.d	odd	2	1
704.3.h.g		6	88.g	even	2	1
704.3.h.h		6	8.b	even	2	1
704.3.h.h		6	88.b	odd	2	1
792.3.j.a		6	3.b	odd	2	1
792.3.j.a		6	33.d	even	2	1
1584.3.j.k		6	12.b	even	2	1
1584.3.j.k		6	132.d	odd	2	1

Hecke kernels

This newform subspace is the entire newspace $S_{3}^{\mathrm{new}}(88, [\chi])$.

Hecke characteristic polynomials

$p$	$F_p(T)$
$2$	$ T^{6} $ T^6
$3$	$ (T^{3} + 2 T^{2} - 13 T - 22)^{2} $ (T^3 + 2T^2 - 13T - 22)^2
$5$	$ (T^{3} - 43 T + 86)^{2} $ (T^3 - 43*T + 86)^2
$7$	$ T^{6} + 232 T^{4} + \cdots + 32768 $ T^6 + 232T^4 + 11520T^2 + 32768
$11$	$ T^{6} - 10 T^{5} + \cdots + 1771561 $ T^6 - 10T^5 + 167T^4 - 2508T^3 + 20207T^2 - 146410*T + 1771561
$13$	$ T^{6} + 904 T^{4} + \cdots + 18096128 $ T^6 + 904T^4 + 232960T^2 + 18096128
$17$	$ T^{6} + 552 T^{4} + \cdots + 131072 $ T^6 + 552T^4 + 65792T^2 + 131072
$19$	$ T^{6} + 1608 T^{4} + \cdots + 13770752 $ T^6 + 1608T^4 + 660992T^2 + 13770752
$23$	$ (T^{3} + 26 T^{2} + \cdots + 514)^{2} $ (T^3 + 26T^2 + 211T + 514)^2
$29$	$ T^{6} + 928 T^{4} + \cdots + 2097152 $ T^6 + 928T^4 + 184320T^2 + 2097152
$31$	$ (T^{3} + 18 T^{2} + \cdots - 12254)^{2} $ (T^3 + 18T^2 - 709T - 12254)^2
$37$	$ (T^{3} + 24 T^{2} + \cdots - 88154)^{2} $ (T^3 + 24T^2 - 2947T - 88154)^2
$41$	$ T^{6} + 3944 T^{4} + \cdots + 375160832 $ T^6 + 3944T^4 + 2360576T^2 + 375160832
$43$	$ T^{6} + \cdots + 2988572672 $ T^6 + 6080T^4 + 8520704T^2 + 2988572672
$47$	$ (T^{3} + 30 T^{2} + \cdots + 73928)^{2} $ (T^3 + 30T^2 - 4516T + 73928)^2
$53$	$ (T^{3} - 54 T^{2} + \cdots - 328)^{2} $ (T^3 - 54T^2 - 1780T - 328)^2
$59$	$ (T^{3} - 118 T^{2} + \cdots + 4714)^{2} $ (T^3 - 118T^2 + 1531T + 4714)^2
$61$	$ T^{6} + \cdots + 110698299392 $ T^6 + 16320T^4 + 80282624T^2 + 110698299392
$67$	$ (T^{3} + 146 T^{2} + \cdots + 31306)^{2} $ (T^3 + 146T^2 + 4339T + 31306)^2
$71$	$ (T^{3} - 150 T^{2} + \cdots - 3998)^{2} $ (T^3 - 150T^2 + 1523T - 3998)^2
$73$	$ T^{6} + 8808 T^{4} + \cdots + 528515072 $ T^6 + 8808T^4 + 13935872T^2 + 528515072
$79$	$ T^{6} + 20608 T^{4} + \cdots + 33554432 $ T^6 + 20608T^4 + 18743296T^2 + 33554432
$83$	$ T^{6} + \cdots + 560545660928 $ T^6 + 26528T^4 + 217567232T^2 + 560545660928
$89$	$ (T^{3} - 192 T^{2} + \cdots - 80082)^{2} $ (T^3 - 192T^2 + 9837T - 80082)^2
$97$	$ (T^{3} - 200 T^{2} + \cdots + 175694)^{2} $ (T^3 - 200T^2 + 7557T + 175694)^2
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Elliptic curves over $\Q$
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Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 88 = 2^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	88.h (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\( q + ( - \beta_1 - 1) q^{3} + \beta_{3} q^{5} - \beta_{2} q^{7} + (\beta_{3} + 1) q^{9} + (\beta_{4} + \beta_{3} - \beta_{2} + \cdots + 2) q^{11} + (\beta_{5} - 2 \beta_{4} + \beta_{2}) q^{13} + ( - 2 \beta_{3} - 3 \beta_1 - 1) q^{15}+ \cdots + (4 \beta_{5} - \beta_{4} + \beta_{3} + \cdots + 33) q^{99}+O(q^{100}) \) q + (-b1 - 1) * q^3 + b3 * q^5 - b2 * q^7 + (b3 + 1) * q^9 + (b4 + b3 - b2 + b1 + 2) * q^11 + (b5 - 2b4 + b2) q^13 + (-2b3 - 3b1 - 1) * q^15 + (-b5 - b2) * q^17 + (2b4 + b2) q^19 + (-b5 + 3b2) q^21 + (-b1 - 9) * q^23 + (-3b3 + 4b1 + 5) * q^25 + (-2b3 + 5b1 + 7) * q^27 - 2b4 q^29 + (-4b3 + 3b1 - 5) * q^31 + (b5 + 2b4 - 3b3 + b2 - 4b1 - 12) q^33 + (4b5 - 4b4 + b2) * q^35 + (5b3 + 12b1 - 4) * q^37 + (-4b5 + b2) q^39 + (b5 + 4b4 - 3b2) * q^41 + (-4b5 + 2b4 - 2b2) q^43 + (-2b3 + 4b1 + 30) * q^45 + (10b3 - 6b1 - 12) * q^47 + (-4b3 - 20b1 - 35) * q^49 + (-4b4 + 3b2) * q^51 + (4b3 - 12b1 + 14) * q^53 + (4b5 - 2b4 + 3b2 + 7b1 + 31) * q^55 + (5b5 + 4b4 - 7b2) q^57 + (4b3 - 13b1 + 35) * q^59 + (-6b5 - 2b4 + 2b2) q^61 + (4b5 - 4b4) * q^63 + (-5b5 + 4b4 - 13b2) q^65 + (8b3 - b1 - 49) q^67 + (b3 + 8b1 + 18) q^69 + (8b3 + 15b1 + 55) * q^71 + (-b5 + 4b4 + 3b2) * q^73 + (2b3 + 8b1 - 38) * q^75 + (5b5 - 4b4 + 4b3 - 3b2 - 12b1 - 44) q^77 + (4b5 + 8b4 - 4b2) q^79 + (-10b3 + 4b1 - 59) * q^81 + (-8b5 - 2b4) * q^83 + (5b5 - 8b4 + 9b2) q^85 + (-4b5 - 4b4 + 4b2) q^87 + (-3b3 - 12b1 + 60) * q^89 + (-28b3 - 12b1 - 12) * q^91 + (5b3 + 20b1 - 18) * q^93 + (-4b5 + 8b4 + 3b2) q^95 + (-7b3 + 16b1 + 72) * q^97 + (4b5 - b4 + b3 + 2b2 + 8b1 + 33) q^99
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q - 4 q^{3} + 6 q^{9} + 10 q^{11} - 52 q^{23} + 22 q^{25} + 32 q^{27} - 36 q^{31} - 64 q^{33} - 48 q^{37} + 172 q^{45} - 60 q^{47} - 170 q^{49} + 108 q^{53} + 172 q^{55} + 236 q^{59} - 292 q^{67} + 92 q^{69}+ \cdots + 182 q^{99}+O(q^{100}) \) 6 * q - 4 * q^3 + 6 * q^9 + 10 * q^11 - 52 * q^23 + 22 * q^25 + 32 * q^27 - 36 * q^31 - 64 * q^33 - 48 * q^37 + 172 * q^45 - 60 * q^47 - 170 * q^49 + 108 * q^53 + 172 * q^55 + 236 * q^59 - 292 * q^67 + 92 * q^69 + 300 * q^71 - 244 * q^75 - 240 * q^77 - 362 * q^81 + 384 * q^89 - 48 * q^91 - 148 * q^93 + 400 * q^97 + 182 * q^99

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